A note on the multicolour version of the Erdős-Hajnal conjecture

IF 0.7 3区数学 Q2 MATHEMATICS

Discrete Mathematics Pub Date : 2025-02-13 DOI:10.1016/j.disc.2025.114437

Maria Axenovich, Lea Weber

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引用次数: 0

Abstract

Informally, the multicolour version of the Erdős-Hajnal conjecture (shortly EH-conjecture) asserts that if a sufficiently large host clique on n vertices is edge-coloured avoiding a copy of some fixed edge-coloured clique, then there is a large homogeneous set of size

n^{β}

for some positive β, where a set of vertices is homogeneous if it does not induce all the colours. This conjecture, if true, claims that imposing local conditions on edge-partitions of cliques results in a global structural consequence such as a large homogeneous set, a set avoiding all edges of some part. While this conjecture attracted a lot of attention, it is still open even for two colours.

In this note, we reduce the multicolour version of the EH-conjecture to the case when the number of colours used in a host clique is either the same as in the forbidden pattern or one more. We exhibit a non-monotonicity behaviour of homogeneous sets in coloured cliques with forbidden patterns by showing that allowing an extra colour in the host graph could actually decrease the size of a largest homogeneous set.

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来源期刊

Discrete Mathematics 数学-数学

CiteScore

1.50

自引率

12.50%

发文量

424

审稿时长

6 months

期刊介绍： Discrete Mathematics provides a common forum for significant research in many areas of discrete mathematics and combinatorics. Among the fields covered by Discrete Mathematics are graph and hypergraph theory, enumeration, coding theory, block designs, the combinatorics of partially ordered sets, extremal set theory, matroid theory, algebraic combinatorics, discrete geometry, matrices, and discrete probability theory. Items in the journal include research articles (Contributions or Notes, depending on length) and survey/expository articles (Perspectives). Efforts are made to process the submission of Notes (short articles) quickly. The Perspectives section features expository articles accessible to a broad audience that cast new light or present unifying points of view on well-known or insufficiently-known topics.